Wednesday, February 10, 2016

NCTM Essential Mathematics Teaching Practice #3—Use and Connect Mathematical Representations

Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving.
It was within my first week of professional learning as a math interventionist that I realized what I had done.  Well not what I had done, but had neglected to do, that was very likely the root of my students’ mathematical struggles over the years.  I wasn’t alone as you could hear the murmur of teachers across the state remorsefully apologizing to the years of students that they had wronged.  
What, you may ask, can move a competent teacher leader to want to fire herself from her charge?  The 3 aspects of number.  Such a tiny piece of information and yet so powerful.  As I learned through Math Recovery, behind every number are 3 dimensions, verbal (what we say “three”), quantity (the amount that we see “* * *”), and the symbol (“3”).  To be proficient, math students need a complete understanding of the numbers around them.  For many that are struggling, one or more of these dimensions are missing or flawed.  As an educator, I had failed to provide my students with a balance of these 3 aspects in my instructional program and their shallow interpretations were showing in peculiar ways.
I could still recall my Melanie who would write the strangest answers on her math assignments.  One day while students were “making 10” to add, she came to me with her paper.  “Mrs. Rowland, I just don’t get it.”  On her paper was written 9 + 6.  I explained, “All you have to do is take one from the six and give it to the nine.  That makes it ten plus five.  You know that one.” She stood there a few moments longer before informing me that there was no one in the problem, just a nine and a six.
Truly, Melanie wasn’t the only student that suffered.  Even my quick, eager learners were limited in their ability to think deeply about problems.  Through my greatest efforts of timed tests and bare number tasks I had created a league of the fastest counters known to man.  These students, though quick, were still counting by one for the majority of the problems they encountered.  I had done very little to help them think deeply about the mathematics they were learning.
According to NCTM, having students use mathematical representations (which have significant overlap with the 3 aspects of number), and helping them make connections between them, is critical in getting students to think deeply about mathematics.  These representations include visual, symbolic, verbal, contextual, and physical, each one providing a new dimension in which to view mathematical concepts. The diagram shown here illustrates conceptual networks developed when using various representations (NCTM, 2014, p. 25).

As students move between these representations their understanding of math concepts deepen.  These representations provide opportunities for students to look for structure, analyze essential elements of mathematical ideas, and communicate their thinking.  As noted by the NCTM (2014, p. 25): According to the National Research Council (2001), “Because of the abstract nature of mathematics, people have access to mathematical ideas only through the representations of those ideas.”  When students have strong connections to these representations, their ability to solve problems is greater as they can move with ease between them to make sense of the problem.  Connectedness of representations is not only good for our students, it’s essential!  
In my intervention classroom, the opportunities that students had to move between these representations helped them to understand mathematics in an authentic way.  The more responsibility for making those connections that I could give to the students, the stronger and more lasting their understanding.  We used mathematical tools to explore ideas and visual models to record our thinking.  These visual models provided opportunities for students to compare, defend, and debate strategies and ideas.  Intervention students benefited as well from the additional time and space to look deeper at the algorithms that they had been taught either in the classroom or at home.  These connections allowed them to explore the validity of these algorithms and reflect on their understanding of the process giving students the perfect climate to be active in their own learning.  
Ready to give your lessons a make-over?  Here are some of my top tips for creating connections in the classroom!
  1. Teach concepts in context.  Let students ponder mathematical ideas in the real world.  Traditional textbook series start with the math then end with the application.  Don’t be afraid to flip it!  Teachers in middle and high school grades can explore some high interest projects at the website  Students can engage in mathematics around how to become more popular on Instagram or to determine exactly how much your pizza is costing you.  For younger grades I recommend Marilyn Burns Math Solutions as an excellent resource for teaching in context.  Here is a link to her blog with tips on getting started:

  1. Know the representations that fit best with the concept that you are about to teach.  Some great places to look are John Van DeWalle’s Teaching Student Centered Mathematics,, and the Kentucky Numeracy Project.  You can also get a deeper understanding of the intent of the standards and their representations from the coherence map at  Be prepared to allow students to explore the mathematical ideas using these representations, not just introduce them and move on.  If you teach from a textbook series you may want to ask yourself who is reasoning with the models.  If the answer is you or the book you may need to modify the lesson to put the responsibility on the students.

Be aware that physical models and manipulatives are great for allowing students to reason through mathematical ideas at all grade levels.  Make sure to allow students to use the concrete representations for exploration and mathematical reasoning.  If the teacher is simply modeling how one would use the model to get an answer, students may interpret that the learning goal is just that, copy the teacher.  

  1. As your students share their ideas and strategies in classroom discussion, model their thinking on the board using the best representation that matches their thinking.  This allows a natural place for students to engage in the model.  Allow students time to discuss and debate using the model or representation as the backdrop.  Be prepared to shift models if the strategy shifts.  You can help prepare students by having them map out their thinking prior to sharing.  Encourage students to record their thinking, including labeling their work, so that they are able to share their thoughts with a peer.  

  1. Make “Math Talk” a regular part of daily math instruction.  Provide opportunities for students to talk through the math they are learning.  Math talk goes beyond whole group conversation.  Think about natural opportunities in the lesson for students to clarify their thinking with a peer.  Many teachers use math workstations or guided math centers for this purpose.  Whatever your structure, you want to maximize engagement!  Think Pair Share allows many students to engage in math talk at a time.  

  1. Be prepared to model your thinking for students, sharing what you notice and what you wonder about.  Allow students time in inquiry mode.  Bring them together to discuss strategies and ideas.  Your job is to formalize the math for your students.  Keep in mind your big ideas and goals for your unit.

  1. Review your lesson plans for opportunities to connect the 5 types of representations, thinking about how to ensure students will interact with multiple representations.  This is a great job for math PLCs to work on together!

  1. Give your independent practice work a check-up!  How many connections do students need to practice to complete their independent work?  There is more power for our students to complete 1 carefully chosen task in multiple ways, using various representations, than lengthy assignments with multiple problems and limited connections.  Less can be more!  Better yet, allow students to select their problem from a generated set of tasks and demonstrate their thinking in multiple ways!  

  1. Use these representations to provide feedback that moves students deeper into the concept!   Ask students questions to illustrate a new connection.  If the task was bare number ask, “Can you write a story problem that would match this number sentence?”  
Nothing can replace the power of supporting students to make connections using various representations.  We would love to hear from you!  What great ideas do you have for using representations and connecting mathematical ideas?  Tell us what you think!  Share with us on Twitter @kycenterformath #KCMTalks or on Facebook at Kentucky Center for Mathematics.    

National Council of Teachers of Mathematics. (2014). Principles to actions; ensuring mathematical success for all. NCTM, Reston, VA.